Question

question 1 (multiple choice worth 1 points) (06.04 mc) if a triangular cross - section with coordinates at (1, 1), (1, 4) and (3, 1) is rotated about the line x = 1, what will be the resulting three - dimensional object?
options:

• cone
• cylinder
• sphere
• pyramid

Explanation:

1. First, analyze the triangle's coordinates: \((1,1)\), \((1,4)\), and \((3,1)\). When rotated about the line \(x = 1\):

• The side between \((1,1)\) and \((1,4)\) is vertical (since \(x\)-coordinate is constant \(1\)) and has length \(4 - 1=3\).
• The side between \((1,1)\) and \((3,1)\) is horizontal (since \(y\)-coordinate is constant \(1\)) and has length \(3 - 1 = 2\).
• The third side is the hypotenuse.

1. When a right - triangle (this triangle is a right - triangle with legs along \(x = 1\) (vertical leg) and horizontal line \(y = 1\)) is rotated about one of its legs (here, the vertical leg \(x = 1\)):

• The horizontal leg (length \(2\)) will trace a circular base with radius \(r = 2\), and the vertical leg (length \(3\)) will be the height \(h=3\) of the resulting solid.
• The solid formed by rotating a right - triangle about one of its legs is a cone. A sphere is formed by rotating a semicircle, a cylinder is formed by rotating a rectangle, and a pyramid is not formed by rotating a triangle about a line (it is a polyhedron with a polygonal base and triangular faces).

Answer:

A. Cone